Problem: Let $f(x) = 4x + c$ and $g(x) = cx + 2.$  If $f(g(x)) = 12x + d,$ then find $d.$
Solution: We have that
\[f(g(x)) = f(cx + 2) = 4(cx + 2) + c = 4cx + c + 8 = 12x + d.\]Matching coefficients, we get $4c = 12$ and $d = c + 8,$ so $c = 3,$ and $d = 3 + 8 = \boxed{11}.$